PTT 201/4 THERMODYNAMICS
SEM 1 (2013/2014)
1

• Develop the equilibrium criterion for reacting systems based on the second law of thermodynamics.
• Develop a general criterion for chemical equilibrium applicable to any reacting system based on minimizing the Gibbs function for the system.
• Define and evaluate the chemical equilibrium constant.
• Apply the general criterion for chemical equilibrium analysis to reacting idealgas mixtures.
• Establish the phase equilibrium for nonreacting systems in terms of the specific Gibbs function of the phases of a pure substance.
• Apply the Gibbs phase rule to determine the number of independent variables associated with a multicomponent, multiphase system.
• Apply Henry’s law and Raoult’s law for gases dissolved in liquids.
2

We may know the temperature, pressure, and composition (thus the state) of a system but we are unable to predict whether the system is in chemical equilibrium.
The chemical composition of the mixture does not change unless the temperature or the pressure of the mixture is changed. That is, a reacting mixture, in general, has different equilibrium compositions at different pressures and temperatures.
A reaction chamber that contains a mixture of
CO
2
, CO, and O
2 at a specified temperature and pressure.
3

The increase of entropy principle
A chemical reaction in an adiabatic chamber proceeds in the direction of increasing entropy. When the entropy reaches a maximum, the reaction stops.
Equilibrium criteria for a chemical reaction that takes place adiabatically.
4

Combining the first- and the second-law relations for the control mass in the figure
The differential of the Gibbs function (G=H

TS) at constant temperature and pressure
A control mass undergoing a chemical reaction at a specified temperature and pressure.
Criteria for chemical equilibrium for a fixed mass at a specified T and P.
from the two equations
A chemical reaction at a specified T and P proceeds in the direction of a decreasing Gibbs function. The reaction stops and chemical equilibrium is established when the Gibbs function attains a minimum value. Therefore, the criterion for chemical equilibrium is
5

To obtain a relation for chemical equilibrium in terms of the properties of the individual components, we consider a mixture of four chemical components A, B, C, and D that exist in equilibrium at a specified T and P.
From the equilibrium criterion
Criterion for chemical equilibrium
An infinitesimal reaction in a chamber at constant temperature and pressure.
To find a relation between the dN’s, we write the corresponding stoichiometric
(theoretical) reaction the

’s are the stoichiometric coefficients
 is the proportionality constant. Substituting,
The changes in the number of moles of the components during a chemical reaction are proportional to the stoichiometric coefficients regardless of the extent of the reaction.
6

7

Now we define the equilibrium constant K
P for the chemical equilibrium of ideal-gas mixtures as
Substituting into Eq. 16–12 and rearranging, we obtain
Therefore, the equilibrium constant K
P of an ideal-gas mixture at a specified temperature can be determined from a knowledge of the standard-state Gibbs function change at the same temperature. The K
P values for several reactions are given in Table A–28.
Partial pressure in terms of mole numbers
Three equivalent K
P relations for reacting ideal-gas mixtures.
8

9

P
1. The K
P of a reaction depends on temperature only.
It is independent of the pressure of the equilibrium mixture and is not affected by the presence of inert gases. This is because K
P depends on ΔG*(T), which depends on temperature only, and the ΔG*(T) of inert gases is zero. Thus, at a specified temperature the following four reactions have the same K
P value:
2. The K
P
of the reverse reaction is 1/K
P
.
This is easily seen from Eq. 16–13. For example, from Table A–28,
10

3. The larger the K
P
, the more complete the reaction.
If the equilibrium composition consists largely of product gases, the partial pressures of the products are considerably larger than the partial pressures of the reactants, which results in a large value of K
P
In the limiting case of a complete reaction
.
(no leftover reactants in the equilibrium mixture), K
P approaches infinity. Reactions with very small K
P values at a specified temperature can be neglected.
The larger the K
P
, the more complete the reaction.
4. The mixture pressure affects the equilibrium
composition (although it does not affect the equilibrium constant K
P
) . At a specified temperature, the K
P value of the reaction, and thus the right-hand side of Eq. 16–15, remains constant. Therefore, the mole numbers of the reactants and the products must change to counteract any changes in the pressure term.
11

5. The presence of inert gases affects
the equilibrium composition (although it does not affect the equilibrium constant
K
P
).
This can be seen from Eq. 16–15, which involves the term (1/N total
) Δ

, where N total includes inert gases.
6. When the stoichiometric coefficients are doubled, the value of K
For example,
P is squared.
The presence of inert gases does not affect the equilibrium constant, but it does affect the equilibrium composition.
12

7. Free electrons in the equilibrium composition can be treated as an ideal gas. At high temperatures
(usually above 2500 K), gas molecules start to dissociate into unattached atoms, and at even higher temperatures atoms start to lose electrons and ionize.
8. Equilibrium calculations provide information on the equilibrium composition of a reaction, not on the reaction rate.
Sometimes it may even take years to achieve the indicated equilibrium composition.
When the right catalyst is used, the reaction goes to completion rather quickly to the predicted value.
Equilibrium-constant relation for the ionization reaction of hydrogen.
13

14

15

There is a driving force between the two phases of a substance that forces the mass to transform from one phase to another.
The magnitude of this force depends, among other things, on the relative concentrations of the two phases.
A wet T-shirt dries much quicker in dry air than it does in humid air.
In fact, it does not dry at all if the relative humidity of the environment is 100%.
In this case, there is no transformation from the liquid phase to the vapor phase, and the two phases are in phase equilibrium .
The conditions of phase equilibrium change if the temperature or the pressure is changed.
Therefore, we examine phase equilibrium at a specified temperature and pressure.
Wet clothes hung in an open area eventually dry as a result of mass transfer from the liquid phase to the vapor phase.
16

The total Gibbs function of the mixture in the figure is
If during a disturbance a differential amount of liquid dm f evaporates at constant T and P,
At equilibrium, (dG)
T,P of mass, dm g
=  dm f
= 0. Also from the conservation
. Substituting, which must be equal to zero at equilibrium. It yields
The two phases of a pure substance are in equilibrium when each phase has the same value of specific Gibbs function.
A liquid–vapor mixture in equilibrium at a constant temperature and pressure.
At the triple point, the specific Gibbs functions of all three phases are equal to each other. The Gibbs function difference is the driving force for phase change.
17

18

19

A a single-component two-phase system has one independent property, which may be taken to be the temperature or the pressure.
In general, the number of independent variables associated with a multicomponent, multiphase system is given by the Gibbs phase rule :
IV = number of independent variables
C = number of components
PH = number of phases present in equilibrium.
Example 1: At the triple point, PH = 3 and thus IV
= 0. That is, none of the properties of a pure substance at the triple point can be varied.
Example 2: Based on this rule, two independent intensive properties need to be specified to fix the equilibrium state of a pure substance in a single phase
According to the Gibbs phase rule, a singlecomponent, two-phase system can have only one independent variable.
20

Many multiphase systems encountered in practice involve two or more com-ponents. A multicomponent multiphase system at a specified temperature and pressure is in phase equilibrium when there is no driving force between the different phases of each component. Thus, for phase equilibrium, the specific Gibbs function of each component must be the same in all phases
A multicomponent multiphase system is in phase equilibrium when the specific Gibbs function of each component is the same in all phases.
21

the phase equilibrium of two-component systems that involve two phases (liquid and vapor) in equilibrium. For such systems, C = 2, PH = 2, and thus IV = 2. That is, a two-component, two-phase system has two independent variables, and such a system will not be in equilibrium unless two independent intensive properties are fixed.
For the two phases of a two-component system, the mole fraction of a component is different in different phases. On the diagram shown, the vapor line represents the equilibrium composition of the vapor phase at various temperatures, and the liquid line does the same for the liquid phase.
Therefore, once the temperature and pressure
(two independent variables) of a twocomponent, two-phase mixture are specified, the equilibrium composition of each phase can be determined from the phase diagram, which is based on experimental measurements.
Equilibrium diagram for the two-phase mixture of oxygen and nitrogen at 0.1
MPa.
22

It is interesting to note that temperature is a continuous function, but mole fraction
(which is a dimensionless concentration), in general, is not.
The water and air temperatures at the free surface of a lake, for example, are always the same. The mole fractions of air on the two sides of a water–air interface, however, are obviously very different (in fact, the mole fraction of air in water is close to zero).
Likewise, the mole fractions of water on the two sides of a water–air interface are also different even when air is saturated. Therefore, when specifying mole fractions in two-phase mixtures, we need to clearly specify the intended phase.
Unlike temperature, the mole fraction of species on the two sides of a liquid–gas (or solid–gas or solid–liquid) interface are usually not the same.
23

Many processes involve the absorption of a gas into a liquid.
1. Gases weakly soluble in liquids
Most gases are weakly soluble in liquids (such as air in water), and for such dilute solutions the mole fractions of a species i in the gas and liquid phases at the interface are observed to be proportional to each other.
That is, y
i,gas side
 y
i,liquid side
/P for ideal-gas mixtures. or P
i,gas side

Py
i,liquid side since y i
= P i
This is known as the Henry’s law and is expressed as where H is the Henry’s constant, which is the product of the total pressure of the gas mixture and the proportionality constant.
For a given species, it is a function of temperature only and is practically independent of pressure for pressures under about 5 atm.
24

From the table and Henry’s law:
1.
The concentration of a gas dissolved in a liquid is inversely proportional to Henry’s constant.
2.
The Henry’s constant increases (and thus the fraction of a dissolved gas in the liquid decreases) with increasing temperature.
3.
The concentration of a gas dissolved in a liquid is proportional to the partial pressure of the gas.
Therefore, the amount of gas dissolved in a liquid can be increased by increasing the pressure of the gas (e.g., the carbonation of soft drinks with CO
2 gas).
Dissolved gases in a liquid can be driven off by heating the liquid.
25

2. Gases highly soluble in liquids
When the gas is highly soluble in the liquid (or solid), such as ammonia in water, the linear relationship of Henry’s law does not apply, and the mole fraction of a gas dissolved in the liquid (or solid) is usually expressed as a function of the partial pressure of the gas in the gas phase and the temperature.
An approximate relation in this case for the mole fractions of a species on the liquid and gas sides of the interface is given by Raoult’s law as
P
i,sat
(T) = the saturation pressure of the species i at the interface temperature
P total
= the total pressure on the gas phase side
The molar density of the gas species i in the solid at the interface is proportional to the partial pressure of the species i in the gas on the gas side of the interface and is expressed as solubility
The product of solubility of a gas and the diffusion coefficient of the gas in a solid is referred to as the permeability , which is a measure of the ability of the gas to penetrate a solid. Permeability is inversely proportional to thickness.
26

27

28

•
•
•
P
•
–
–
–
29


30